Proof of the absence of local conserved quantities in the Holstein model

Absence of local conserved quantities, or \textit{nonintegrability}, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron--phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein--Hubbard model. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.